Consider a particle undergoing Brownian motion in $\mathbb{R}^n$, starting at the origin, and let $B(t)$ denote its position at time $t$. Let $X$ be an arbitrary subset of $\mathbb{R}^n$. I am trying to understand what properties $X$ needs to have so that the probability of the Brownian particle striking $X$ within time $t$ is non-zero, that is, $\mathbb{P}(B(s) \in X, s \leq t) \neq 0$. It seems to me that a sufficient condition on $X$ could be that it contains a subset $Y$ which is homeomorphic to $\mathbb{R}^{n - 1}$. Is this necessary too? Is there a more relaxed sufficient condition? Thanks in advance.